We shall follow closely Meinrenken's ideas to construct an Euler-like multiplicative vector field for \((G,G_\mathcal {O})\)  combined with his application regarding the proof of the Morse-Bott lemma mentioned right above. As the groupoid orbit \(\mathcal {O}\)  is non-degenerate for \(f\)  there exists an Euler-like vector field \(Y\)  for \((M,\mathcal {O})\)  such that \(\mathcal {L}_Yf=2f\)  near \(\mathcal {O}\) . Since \(s\)  is a surjective submersion there is a vector field \(X\)  on \(G\)  such that \(X\)  is \(s\) -related with \(Y\) . The fact that \(\overline{ds}:\nu (G_{\mathcal {O}})\rightarrow \nu (\mathcal {O})\)  is a fiberwise isomorphism implies that \(X\)  is Euler-like for \((G, G_\mathcal {O})\) . Furthermore, \(\mathcal {L}_XF=2F\)  near \(G_\mathcal {O}\)  because \(F=s^\ast f\)  and \(X\)  is \(s\) -related with \(Y\) .
Let us now consider a proper Haar measure system \(\lbrace \mu ^x\rbrace \)  for \(G\rightrightarrows M\) . By following results due to Crainic and Struchiner in [1]} we can construct a multiplicative vector field \(\overline{X}:G\rightarrow TG\)  by taking the average with respect to \(\lbrace \mu ^x\rbrace \) :
\(\overline{X}(g)=\int _{a\in t^{-1}(s(g))}dm_{(ga,a^{-1})}(X(ag),di_a(X_a))\mu (a).\) 
This vector field is \(s\) -related with the vector field \(\overline{Y}\)  on \(M\)  defined as
\(\overline{Y}(x)=\int _{a\in t^{-1}(x)}dt_a(X_a)\mu (a).\) 
Meinrenken showed that both \(\overline{X}\)  and \(\overline{Y}\)  are Euler-like vector fields, thus proving that \(G\rightrightarrows M\)  is weakly linearizable around \(G_{\mathcal {O}}\rightrightarrows \mathcal {O}\)  (see [2]}). Therefore, our result will follow once we prove that \(\mathcal {L}_{\overline{Y}}f=2f\)  and \(\mathcal {L}_{\overline{X}}F=2F\)  near \(\mathcal {O}\)  and \(G_\mathcal {O}\) , respectively. Indeed, on the one hand it follows that
\(\mathcal {L}_{\overline{Y}}f(x) &=& \int _{a\in t^{-1}(x)}d(f\circ t)_a(X_a)\mu (a)=\int _{a\in t^{-1}(x)}df_{s(a)}(Y(s(a))\mu (a)\\&=& 2 \int _{a\in t^{-1}(x)}(f\circ s)(a)\mu (a)=\int _{a\in t^{-1}(x)}(f\circ t)(a)\mu (a)=2f(x),\) 
since \(X\)  is \(s\) -related with \(Y\)  and \(f\)  is basic. On the other hand, as \(\overline{X}\)  and \(\overline{Y}\)  are \(s\) -related then from the previous computation we trivially obtain that \(\mathcal {L}_{\overline{X}}F=2F\)  near \(G_\mathcal {O}\) , as desired.
